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The dynamics of probability density functions has been extensively studied in science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics that can be formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. The goal of this paper is to parallel the success of techniques using functional inequalities, for dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences as energy functionals. Such dynamics take the form of a nonlocal differential equation, for which existing analysis critically relies on using the explicit solution formula in special cases. We provide a comprehensive study on functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of the obtained functional inequalities is that they do not depend on the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posed) is uniform across general target distributions, making them potentially desirable dynamics for posterior sampling applications in Bayesian inference., Comment: 38 pages

We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. The essential challenge lies in the fact that the weak dissipation and randomness interact in the evolution.

In this paper, we solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our weak solution equals to Lions' weak solution, which in particular implies the uniqueness of Lions' weak solution. We also extend a celebrated result by Fujita and Kato on the 3-D incompressible Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in $\dot H^{1/2}(\mathbb R^3)$}. The proof of the uniqueness is based on a surprising finding that the estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of $\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the uniqueness of the solution., Comment: 23 pages

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $\rho_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.}, Comment: 23 pages

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time., Comment: 56 pages

Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation., Comment: 68 pages, 2 figures

In this paper, we study the long time asymptotic behaviors for solutions to the Chern-Simons-Higgs equation with a pure power defocusing nonlinearity. We obtain quantitative inverse polynomial time decay for the potential energy for all data with finite conformal energy. Consequently, the solution decays in time in the pointwise sense for all power. We also show that for sufficiently large power the solution decays as quickly as linear waves. Key ingredients for the proof include vector field method, conformal compactification and the geometric bilinear trace theorem for null hypersurface developed by Klainerman-Rodnianski., Comment: 42 pages, comments are welcome!

We show that the 1D Hou-Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We also prove that the profiles satisfy some monotonicity and convexity properties that were unknown before, and we give rigorous estimates on the algebraic decay rates of the profiles in the far field. Our result supplements the previous computer-assisted proof of self-similar finite-time blowup for the Hou-Luo model with finer characterizations of the profiles., Comment: 32 pages, 6 figures

In this paper, we study the long time dynamics of solutions to the defocusing semilinear wave equation $\Box_g\phi=|\phi|^{p-1}\phi$ on the Schwarzschild black hole spacetimes. For $\frac{1+\sqrt{17}}{2}

We show that the $a$-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an $a$-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of $a$ and also explains previous numerical observations for a wide range of $a$., Comment: 48 pages, 10 figures

In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and $\mathring{\omega}\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$) and a dominant condition. As an important application, we prove the existence of weak solution when $\mathring{\omega}$ is a Radon measure on $\mathbb T$ with $m$-fold symmetry, which is related to the vortex sheet solution., Comment: 67 pages, 1 figure

The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quantities of the moving string satisfy extremum principles and decay estimates. As a result, we can prove that the 2-D Peskin problem admits a unique global solution when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed., Comment: A new Remark 3.1 is added to v2

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, this is the first global well-posedness result for the 2-D inhomogeneous incompressible Euler equations., Comment: 64 pages

It has been shown in [Yang-Yu 2019] that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space $\mathbb{R}^{1+3}$ decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of $(r\underline{\alpha}, r\phi)$ along the out going null geodesics. In this paper, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem for the MKG system with general large data by using gauge invariant vector field method. We in particular extend the small data result of He in \cite{MR4299134} to a class of general large data., Comment: 43pages,5 figures

Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since Hall, M\"uller and Wang (2006)'s result for a fixed number of eigenfunction estimates. In this work, we aim to establish a unified theory for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the $\mathcal{L}^2$ and supremum norms, and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. Our results provide insight into the phenomenon when the $\mathcal{L}^{2}$ bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed, and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. We also develop a double truncation technique to handle the uniform convergence of estimated covariance and eigenfunctions. The technical arguments in this work are useful for handling the perturbation series with noisy and discretely observed functional data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.

We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator., Comment: 33 pages

In this paper, we study the extremal process of the maxima of all the largest eigenvalues of principal minors of the classical Gaussian orthogonal ensemble (GOE). We prove that the fluctuation of the maxima is given by the Gumbel distribution in the limit. We also derive the limiting joint distribution of the maxima and the corresponding eigenvector, which implies that these two random variables are asymptotically independent.

We study solutions to the Yang-Mills-Higgs equations on the maximal Cauchy development of the data given on a ball of radius $R$ in $\mathbb{R}^3$. The energy of the data could be infinite and the solution grows at most inverse polynomially in $R-t$ as $t\rightarrow R$. As applications, we derive pointwise decay estimates for Yang-Mills-Higgs fields in the future of a hyperboloid or in the Minkowski space $\mathbb{R}^{1+3}$ for data bounded in the weighted energy space with weights $|x|^{1+\epsilon}$. Moreover, for the abelian case of Maxwell-Klein-Gordon system, we extend the small data result of Lindblad and Sterbenz to general large data (under same assumptions but without any smallness). The proof is gauge independent and it is based on the framework of Eardley and Moncrief together with the geometric Kirchhoff-Sobolev parametrix constructed by Klainerman and Rodnianski. The new ingredient is a class of weighted energy estimates through backward light cones adapted to the initial data., Comment: 40 pages. All comments are welcome!

By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity for all $1

In this paper, we study the number of traveling wave families near a shear flow under the influence of Coriolis force, where the traveling speeds lie outside the range of the flow $u$. Under the $\beta$-plane approximation, if the flow $u$ has a critical point at which $u$ attains its minimal (resp. maximal) value, then a unique transitional $\beta$ value exists in the positive (resp. negative) half-line such that the number of traveling wave families near the shear flow changes suddenly from finite to infinite when $\beta$ passes through it. On the other hand, if $u$ has no such critical points, then the number is always finite for positive (resp. negative) $\beta$ values. This is true for general shear flows under mildly technical assumptions, and for a large class of shear flows including a cosine jet $u(y) = {1+\cos(\pi y)\over 2}$ (i.e. the sinus profile) and analytic monotone flows unconditionally. The sudden change of the number of traveling wave families indicates that long time dynamics around the shear flow is much richer than the non-rotating case, where no such traveling wave families exist., Comment: Revision based on referees' suggestions, 47 pages, 3 figures

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$., Comment: 147 pages

This paper is devoted to the study of relativistic Vlasov-Maxwell system in three space dimension. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimate. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with rate $(1+|x|+|v|)^{-9-}$. In particular, there is no restriction on the support of the initial data., Comment: 29 pages, comments are welcome!

This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality., Comment: 21 pages, comments are welcome!

In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+\sqrt{8}$. Moreover combined with Br\'{e}zis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-\frac{1}{2}}$ when $p>\frac{11}{3}$ and with rate $t^{ -\frac{p-1}{8}+\epsilon }$ for all $12\sqrt{5}-1$., Comment: 28 pages. Comments are welcome

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This is a long-standing problem since the experiments of Reynolds in 1883. Our work lays a foundation for the theoretical analysis of hydrodynamic stability of pipe flow, which is one of the oldest yet unsolved problems of fundamental fluid dynamics., Comment: Any comments are welcome

We will prove the Berry-Esseen theorem for the number counting function of the circular $\beta$-ensemble (C$\beta$E), which will imply the central limit theorem for the number of points in arcs of the unit circle in mesoscopic and macroscopic scales. We will prove the main result by estimating the characteristic functions of the Pr\"ufer phases and the number counting function, which will imply the uniform upper and lower bounds of their variance. We also show that the similar results hold for the Sine$_\beta$ process. As a direct application of the uniform variance bound, we can prove the normality of the linear statistics when the test function $f(\theta)\in W^{1,p}(S^1)$ for some $p\in(1,+\infty)$., Comment: the title has been changed for this version, which was 'Normality of Circular $\beta$-ensemble'

In this article, we study the smallest gaps of the Gaussian orthogonal ensemble. The main result is that the smallest gaps, after normalized by $n$, will tend to a Poisson distribution, and the limiting density of the $k$-th normalized smallest gaps is $ 2{}x^{2k-1}e^{-x^{2}}/(k-1)!$.

In this paper, we first present a Gearhardt-Pr\"uss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales., Comment: 12 pages

In this paper, we study the linear inviscid damping for the linearized $\beta$-plane equation around shear flows. We develop a new method to give the explicit decay rate of the velocity for a class of monotone shear flows. This method is based on the space-time estimate and the vector field method in sprit of the wave equation. For general shear flows including the Sinus flow, we also prove the linear damping by establishing the limiting absorption principle, which is based on the compactness method introduced by Wei-Zhang-Zhao in \cite{WZZ2}. The main difficulty is that the Rayleigh-Kuo equation has more singular points due to the Coriolis effects so that the compactness argument becomes more involved and delicate., Comment: 41 pages

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow $(y,0)$ at large Reynolds number $Re$ in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity $v_0$ satisfies $\|v_0-(y, 0)\|_{H^2}\le cRe^{-\frac 12}$ for some small $c$ independent of $Re$, then the solution of the 2-D Navier-Stokes equations remains within $O(Re^{-\frac 12})$ of the Couette flow for any time., Comment: 46 pages

In this article, we will study the largest gaps of the classical random matrices of CUE and GUE. The main result is that, in both cases, the rescaling largest gaps will converge to the Poisson point processes, and the limiting densities are given by the Gumbel distributions.

In \cite{FTD1}, we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of \emph{law of large numbers} in probability theory; in \cite{FTD2}, we proved that the linear statistic of eigenvalues satisfies the \emph{central limit theorem}. In this article, we continue to study another important theorem in probability theory\,-- the \emph{concentration of measure theorem}, especially for the Gaussian SYK model. We will prove a \emph{large deviation principle} (LDP) for the normalized empirical measure of eigenvalues when $q_n=2$, in which case the eigenvalues can be expressed in term of these of Gaussian random antisymmetric matrices. Such LDP result has its own independent interest in random matrix theory. For general $q_n\geq 3$, we can not prove the LDP, we will prove a concentration of measure theorem by estimating the Lipschitz norm of the Gaussian SYK model.

In our previous paper \cite{FTD1}, we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistic of eigenvalues and compute its variance.

In this article, we study the smallest gaps of the log-gas $\beta$-ensemble on the unit circle (C$\beta$E), where $\beta$ is any positive integer. The main result is that the smallest gaps, after being normalized by $n^{\frac {\beta+2}{\beta+1}}$, will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the $k$-th smallest gap, which is proportional to $x^{k(\beta+1)-1}e^{-x^{\beta+1}}$. In particular, the result applies to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest., Comment: to appear in Annals of Probability in the forthcoming issue

Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\cdots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito \cite{dB90} to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Chern's conjecture., Comment: 16 pages

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures., Comment: 49 pages

This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.

In this paper, we establish the pseudospectral bound for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [LWZ], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies $\big\| U_0-\big(k_f^{-2}\sin(k_fy),0,0\big)\big\|_{H^2}\le c\nu^{\frac{7}{4}}$ ($\nu$ the viscosity coefficient) and $k_f\in (0,1)$, then the solution does not transition away from the Kolmogorov flow., Comment: 66 pages

In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of the Kolmogorov flow with the size of $\nu^{\frac 23+}$, here $\nu$ is the viscosity coefficient., Comment: 92 pages